Upper bounds for the entropy in the cusp for one-parameter diagonal flows on $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$

Abstract

We give explicit upper bounds for the entropy in the cusp for one-parameter diagonal flows on $SL_d(\mathbb{R})/SL_{d}(\mathbb{Z})$. These results include bounds for the entropy of the cusp as a whole, as well as for the cusp regions corresponding to either the maximal parabolic subgroups of $SL_{d}$, or the minimal (Borel) parabolic subgroup. To do so, we use a method which involves choosing an auxiliary linear functional on the Lie algebra of the Cartan group, specifically tailored to each of the bounds we are interested in. In a follow-up paper we prove that the upper bounds we obtain in this work are tight.